# Hecke algebra

Let $f$ be a modular form for $\Gamma$ a congruence subgroup of $\textrm{SL}_{2}(\mathbb{Z})$.

 $f(z)=\underset{n=0}{\overset{\infty}{\sum}}a_{n}q^{n}$ (1)

where $q=e^{2i\pi\tau}$.

For $m\in\mathbb{N}$, let $T_{m}f(z)=\underset{n=0}{\overset{\infty}{\sum}}b_{n}q^{n}$ with :

 $b_{n}=\underset{d|\gcd(m,n)}{\overset{}{\sum}}d^{k-1}a_{mn/d^{2}}$ (2)

In particular, for $p$ a prime, $T_{p}f(z)=\underset{n=0}{\overset{\infty}{\sum}}b_{n}q^{n}$ with;

 $b_{n}=a_{pn}+p^{k-1}a_{n/p}$ (3)

where $a_{n/p}=0$ if $n$ is not divisible by $p$.

The operator $T_{n}$ is a linear operator on the space of modular forms called a Hecke operator.

The Hecke operators leave the space of modular forms and cusp forms invariant and turn out to be self-adjoint for a scalar product called the Petersson scalar product. In particular they have real eigenvalues. Hecke operators also satisfy multiplicative properties that are best summarized by the formal identity:

 $\underset{n=1}{\overset{\infty}{\sum}}T_{n}n^{-s}=\underset{p}{\prod}(1-T_{p}p% ^{-s}+p^{k-1-2s})^{-1}$ (4)

That equation in particular implies that $T_{m}T_{n}=T_{n}T_{m}$ whenever $\gcd(n,m)=1$.

The set of all Hecke operators is usually denoted $\mathbb{T}$ and is called the Hecke algebra.

## 0.1 Group algebra example

Definition 0.1 Let $G_{lcd}$ be a locally compact totally disconnected group; then the Hecke algebra $\mathcal{H}(G_{lcd})$ of the group $G_{lcd}$ is defined as the convolution algebra of locally constant complex-valued functions on $G_{lcd}$ with compact support.

Such $\mathcal{H}(G)$ algebras play an important role in the theory of decomposition of group representations into tensor products.

Title Hecke algebra HeckeAlgebra 2013-03-22 14:08:13 2013-03-22 14:08:13 olivierfouquetx (2421) olivierfouquetx (2421) 13 olivierfouquetx (2421) Definition msc 11F11 msc 20C08 ModularForms AlgebraicNumberTheory Hecke operator Hecke algebra ${H}(G)$ of the group G